Phase separation and second-order phase transition in the phenomenological model for a Coulomb-frustrated two-dimensional system

We have considered the model of the phase transition of the second order for the Coulomb frustrated 2D charged system. The coupling of the order parameter with the charge was considered as the local temperature. We have found that in such a system, an appearance of the phase-separated state is possible. By numerical simulation, we have obtained different types (“stripes,” “rings,” “snakes”) of phase-separated states and determined the parameter ranges for these states. Thus the system undergoes a series of phase transitions when the temperature decreases. First, the system moves from the homogeneous state with a zero order parameter to the phase-separated state with two phases in one of which the order parameter is zero and, in the other, it is nonzero $\left(\tau >0\right)$. Then a first-order transition occurs to another phase-separated state, in which both phases have different and nonzero values of the order parameter (for $\tau <0)$. Only a further decrease of temperature leads to a transition to a homogeneous ordered state.

Figure 1

The distribution of the order parameter $\mathrm{\Lambda }({\xi }_x=const,{\xi }_y)$ (solid curve) and $\mathrm{\Delta }\rho =\rho ({\xi }_x=const,{\xi }_y)-\overline{\rho }$ (dashed curve) for $\chi =3,A=3$, and $\tau =2.8$ in the inhomogeneous state. The inset shows the distribution of the order parameter in 2D. The figure represents the results of numerical calculations for $\mathcal{N}=512$.

Figure 2

Inhomogeneous states are shown for $A=3,\chi =5$, and $\tau =3.2\left(a\right),2\left(b\right),1\left(c\right),-1\left(d\right),-5\left(e\right),-15\left(f\right),-20\left(g\right),-25\left(h\right)$, respectively. The phase-separated state is stable at $3.2\le \tau \le -27$. The parameter $\tau$ decreases from (a) to (h), corresponding to the decreasing of temperature $T$. All figures represent the results of numerical calculations for $\mathcal{N}=128$. The order parameter changes from ${\mathrm{\Lambda }}_{min}=0$ to ${\mathrm{\Lambda }}_{max}=2.2$ in (a) and (b). The difference between ${\mathrm{\Lambda }}_{max}$ and ${\mathrm{\Lambda }}_{min}$ decreases from (c) to (h).

Figure 3

Inhomogeneous states are shown for $A=2.2,\chi =10$, and $\tau =0.01\left(a\right),-0.1\left(b\right),-0.2\left(c\right),-0.5\left(d\right),-1\left(e\right),-2\left(f\right),-3\left(g\right),-3.7\left(h\right)$, respectively. The inhomogeneous state exists in a more narrow interval of $\tau \left(0.01\le \tau \le -0.37\right)$ for $A=2.2$ than for $A=3$. The decrease of $A$ (increase of Coulomb interaction) leads to the decrease of the interval of $\tau$ where the phase separation is observed. All figures represent the results of numerical calculations for $\mathcal{N}=128$.

Figure 4

The maximum ${\mathrm{\Lambda }}_{max}$ and minimum ${\mathrm{\Lambda }}_{min}$ values of the order parameter in the inhomogeneous states as a function of $\tau$ for different values of $A$. The inset shows the distribution of the order parameter $\mathrm{\Lambda }$ in 2D for $A=2,\tau =-1.25$, and $\xi =10.$ The figure represents the results of numerical calculations for $\mathcal{N}=512$.

Figure 5

The distribution of the order parameter $\mathrm{\Lambda }\left(r\right)$ (solid curve) and $\mathrm{\Delta }\rho =\rho \left(r\right)-\overline{\rho }$ (dashed curve) for $\chi =10,A=2.2$, and negative value $\tau =-3$ in the inhomogeneous state. The direction of $r$ is chosen perpendicular to the strips on the inset. The inset shows the distribution of the order parameter in 2D for this set of parameters. The figure represents the results of numerical calculations for $\mathcal{N}=128$.

Figure 6

(a) The phase diagram of inhomogeneous states in axes $A-\tau$ for $\chi =10$ for which $\mathrm{\Delta }\mathrm{\Phi }<0$. The energy of the inhomogeneous phase-separated state ${\mathrm{\Phi }}_{\mathrm{inhom}}$ is lower than the energy of the homogeneous state ${\mathrm{\Phi }}_{\mathrm{hom}}$ at ${\tau }_2<\tau <{\tau }_1$. $\mathrm{\Delta }\mathrm{\Phi }={\mathrm{\Phi }}_{\mathrm{inhom}}-{\mathrm{\Phi }}_{\mathrm{hom}}$. (b) The phase diagram plotted in the axes $T-1/A$, where $T$ is the temperature. The following parameters were used: $T_c+{\sigma }_1\overline{\rho }=275\mathrm{K},\frac{\tau }{{\alpha }^{\prime }}\sqrt{\frac{{\beta }^3}{\zeta }}=0.3{\mathrm{K}}^{-1}$. Region I is a homogeneous nonmagnetic state with $\mathrm{\Lambda }=0$. Region II is a phase-separated state with zero and nonzero order parameters. Region III is a phase-separated state with nonzero order parameter. Region IV is the homogeneous magnetic state with $\mathrm{\Lambda }\ne 0$. The point at $A=1.8$ is the critical end point. For $A<1.8$ the phase separation is impossible.

Figure 7

Phase diagram in $x-T$ axes. The parameter $A$ is equal to 2.7, and ${\sigma }_1=10$. The area between solid and dash dot lines is a phase-separated state, which corresponds to regions II and III in Fig. 6. Dotted line is the temperature of the phase transition in the absence of a phase-separated state.